3.334 \(\int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^4} \, dx\)

Optimal. Leaf size=81 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)} \]

[Out]

(F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(4*b^(3/2)*d*Log[F]^(3/2))
- F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)*Log[F])

_______________________________________________________________________________________

Rubi [A]  time = 0.170348, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x)^2)/(c + d*x)^4,x]

[Out]

(F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(4*b^(3/2)*d*Log[F]^(3/2))
- F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)*Log[F])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.7481, size = 66, normalized size = 0.81 \[ \frac{\sqrt{\pi } F^{a} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (F \right )}}}{c + d x} \right )}}{4 b^{\frac{3}{2}} d \log{\left (F \right )}^{\frac{3}{2}}} - \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{2 b d \left (c + d x\right ) \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**4,x)

[Out]

sqrt(pi)*F**a*erfi(sqrt(b)*sqrt(log(F))/(c + d*x))/(4*b**(3/2)*d*log(F)**(3/2))
- F**(a + b/(c + d*x)**2)/(2*b*d*(c + d*x)*log(F))

_______________________________________________________________________________________

Mathematica [A]  time = 0.068996, size = 81, normalized size = 1. \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{4 b^{3/2} d \log ^{\frac{3}{2}}(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^4,x]

[Out]

(F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(4*b^(3/2)*d*Log[F]^(3/2))
- F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)*Log[F])

_______________________________________________________________________________________

Maple [A]  time = 0.065, size = 76, normalized size = 0.9 \[ -{\frac{{F}^{a}}{2\, \left ( dx+c \right ) db\ln \left ( F \right ) }{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{{F}^{a}\sqrt{\pi }}{4\,\ln \left ( F \right ) bd}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c)^2)/(d*x+c)^4,x)

[Out]

-1/2/d*F^a*F^(b/(d*x+c)^2)/(d*x+c)/b/ln(F)+1/4/d*F^a/b/ln(F)*Pi^(1/2)/(-b*ln(F))
^(1/2)*erf((-b*ln(F))^(1/2)/(d*x+c))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^4,x, algorithm="maxima")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^4, x)

_______________________________________________________________________________________

Fricas [A]  time = 0.272586, size = 161, normalized size = 1.99 \[ \frac{\sqrt{\pi }{\left (d x + c\right )} F^{a} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) - 2 \, F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{4 \,{\left (b d^{3} x + b c d^{2}\right )} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}} \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^4,x, algorithm="fricas")

[Out]

1/4*(sqrt(pi)*(d*x + c)*F^a*erf(d*sqrt(-b*log(F)/d^2)/(d*x + c)) - 2*F^((a*d^2*x
^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2))*d*sqrt(-b*log(F)/d^2))/((
b*d^3*x + b*c*d^2)*sqrt(-b*log(F)/d^2)*log(F))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**4,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^4,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^4, x)